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arXiv · 2312.15333

Induced subgraph density. VII. The five-vertex path

Tung Nguyen·Alex Scott·Paul Seymour
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Abstract

We prove the Erdős-Hajnal conjecture for the five-vertex path $P_5$; that is, there exists $c>0$ such that every $n$-vertex graph with no induced $P_5$ has a clique or stable set of size at least $n^c$. This completes the verification of the Erdős-Hajnal property of all five-vertex graphs. Our methods combine probabilistic and structural ideas with the iterative sparsification framework introduced in the third and fourth papers in the series.

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Tung Nguyen, Alex Scott, Paul Seymour. 2026-02-23. Induced subgraph density. VII. The five-vertex path. https://arxiv.org/abs/2312.15333

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